In the array below, in how many different ways can we start with the letter A and move from letter to letter (horizontally, vertically, or diagonally), to spell the word "ARCH"?
To start with the letter A and move horizontally, vertically, or diagonally to spell the word "ARCH" in the given array, we can consider the following paths:
Path 1: A -> B -> H -> C
Path 2: A -> B -> C -> H
Path 3: A -> G -> F -> C -> H
Path 4: A -> H -> C -> G
Path 5: A -> H -> G -> F -> C
Therefore, there are 5 different ways we can start with the letter A and move to spell the word "ARCH".
To find out the number of different ways we can spell the word "ARCH" starting from the letter A and moving horizontally, vertically, or diagonally in the given array, we will step-by-step determine all the possible paths.
Array:
A B C D E
F G H I J
K L M N O
P Q R S T
U V W X Y
Step 1: Start at the letter A
A
Step 2: Move to neighboring letters
We can move to the following neighboring letters from A: B, F, G
A - B
|
F
A - B - C
| |
F G
Step 3: Move to neighboring letters
From C, we can move to the following neighboring letters: G, K, L, D
A - B - C - G
| | |
F L D
A - B - C - G
| | |
F K D
Step 4: Move to neighboring letters
From G, there are no valid neighboring letters for the next step.
A - B - C - G
Step 5: Move to neighboring letters
From L, we can move to the following neighboring letters: H, M, Q
A - B - C - G
| | |
F L - H
|
M
A - B - C - G
| | |
F L - H
|
Q
Step 6: Move to neighboring letters
From H, we can move to the following neighboring letters: M, Q, R
A - B - C - G
| | |
F L - H - M
A - B - C - G
| | |
F L - H - Q
A - B - C - G
| | |
F L - H - Q - R
Step 7: Move to neighboring letters
From Q, there are no valid neighboring letters for the next step.
A - B - C - G
| | |
F L - H - Q
Step 8: Move to neighboring letters
From M, we can move to the following neighboring letters: I, N
A - B - C - G
| | |
F L - H - M - I
A - B - C - G
| | |
F L - H - M - N
Step 9: Move to neighboring letters
From I, there are no valid neighboring letters for the next step.
A - B - C - G
| | |
F L - H - M - I
Step 10: Move to neighboring letters
From N, we can move to the following neighboring letters: O, S
A - B - C - G
| | |
F L - H - M - N - O
A - B - C - G
| | |
F L - H - M - N - S
Step 11: Move to neighboring letters
From O, there are no valid neighboring letters for the next step.
A - B - C - G
| | |
F L - H - M - N - O
Step 12: Move to neighboring letters
From S, we can move to the following neighboring letters: T
A - B - C - G
| | |
F L - H - M - N - S - T
So, by traversing through the given array and following all the possible paths, we have found 12 different ways to spell the word "ARCH" starting from the letter A and moving horizontally, vertically, or diagonally.